August 2003 Archives

"Of twelve coins, one is counterfeit and weighs either more or less than all the others. The others weigh the same. With a balance scale, on which one side may be weighed against the other, you are to use only three weighings to determine the counterfeit."

About a year ago I found this cool puzzle in the collection of short stories

"The Palace Thief Stories" by Ethan Canin.

It's in the story "Batorsag and Szerelem" about a math prodigy. I had a fun time solving it (took me a while) and now I have the solution on loose papers lying around on a shelf. So I thought it would be good to transfer it over to a LaTex document. Stay tuned...

I just finished reading a very interesting and entertaining book:

"A Mathematician Reads the Newspaper" by John Allen Paulos.

Each chapter dissects one news story (or a couple of news stories falling in the same category) and it exposes faulty reasoning in it, from wrong interpretation of statistical data to circular logical reasoning to optimistic curve fitting to inappropiate probalistic assumptions etc. Seeing the news stories through the author's eye often leads to conclusions that are very different from the headlines that go with those stories. What surprised me most is how ingrained journalistic conventions with faulty reasoning are and how hard it is to shake yourself free and see through them. This book goes a long way in making you a more skeptical and educated newspaper reader.

In the same vein is this webpage in which philosopher Julian Baggini is collecting faulty reasoning in argumentations.

One other instance of faulty reasoning is worth mentioning here, something Malcolm Gladwell calls "creeping determinism" in an article he wrote for The New Yorker magazine. It describes the faulty reasoning in which events in hindsight seem to have been destined to occur the way they did which invariably raises the question of why haven't the signals that pointed in that way been picked up. Examples in the article are among others the Sep 11 tragedy and the 1973 attack on Israel from the neighboring Arab countries. Why haven't the intelligence agencies "connected the dots" ? The problem is that only in hindsight can the signals that pointed that way be distinguished from the general noise that confronts inteligence agencies nowadays (for example the F.B.I.'s counter terrorism division has sixty-eight thousand outstanding and unassigned leads dating back to 1995).

Maybe it is time to explain the title "kern f" of this blog. I found a great article yesterday describing the attitude of being "enamored with the abstract notion of math and science". I am definitely on the lightweight side of the equation.

You encounter kern f pretty early on if you read any algebra textbook. For group homomorphisms for example it's the subgroup of elements in the domain group that map to the unit element in the image group. It has all sorts of interesting properties, for one thing it is a normal subgroup. And conversely for every normal subgroup of a group there is a group homomorphism for which the subgroup is the kern of that morphism. The factor group built with kern f is isomorph to im f.

I realized I have this "Currently Reading" sidebar section and I'm not making much progress on the book I listed there so I took that sidebar section down. The book is

"Prime Obsession: Bernhard Riemann & the Greatest Unsolved Problem" by John Derbyshire,

in case anyone is interested.

It's still a great book but I have other reading interests right now. I'll pick it up later again. I understand now that it's less pressure for me to finish a book if I mention it in a blog entry than it is if I make it a sidebar section "Currently Reading" fixture. Such are the lessons of a lazy reader.

Solved the prison cells problem.

A whole month has passed since my last entry. I guess I was Spirited Away (had to sneak this link in, just watched the movie last night and it was magical, can't wait to watch it again tonight). It so happened that during my silence a lot of math/cs problems and puzzles have piled up on my desk so with some luck I will be more active in the next weeks. Most of the problems came from the unlikely fact that our group at work got an open req suddenly and we had to do interviews. Boy were we rusty. We haven't had an open req forever. But it lead to us talking about interviews (btw a good discussion on the web about interviews is here) and what questions to ask. This lead to cs puzzles and then to problems which some of the group members knew and liked (not exactly suited for 1-hour interviews though).